# Interdisciplinary Applied Mathematics

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FIGURE 14.1. Two-dimensional meshes used in pFlow simulations of flow in a rough microchannel of aspect ratio 20 : 1. The top plot shows in-phase roughness, and the lower plot shows out-of-phase roughness. The domain is broken up    into    large    elements,    and in    each    element    a spectral    expansion    defined    by

Gauss-Lobatto-Legendre (GLL) points (see detail) is used to represent all fields and data (including the geometry). Here N =12 GLL points are used in each direction.

i

Out-Phase Channel with aspect ratio of 20 : 1

to the corresponding cost for standard finite element methods, but for the same accuracy the spectral element method is more efficient.

Our objective here is to focus on the following two issues, which are very important for microflows, instead of presenting complete discretization details; many different algorithms can be found in (Karniadakis and Sherwin, 1999) for general incompressible and compressible flows. Specifically:

For incompressible microflows, which are viscous-dominated, we present the correct pressure boundary condition to supplement the consistent Poisson equation for the pressure.

For compressible flows we present the characteristic treatment of boundary conditions, which guarantee stability and accuracy.

The efficiency of the overall algorithm is based on time-splitting of the ad-vection and diffusion operators, which are treated with a collocation formulation and Galerkin projection, respectively. The time-splitting is demonstrated here for the one-dimensional linear advection-diffusion equation

Ut + VUX = aUxx,    (14.1)

where V is the constant advection velocity. Splitting the advection and the diffusion terms and discretizing the time derivative, we obtain